# Chapter 4: Probability and Counting Rules

Size: px
Start display at page:

Transcription

1 Chapter 4: Probability and Counting Rules Before we can move from descriptive statistics to inferential statistics, we need to have some understanding of probability: Ch4: Probability and Counting Rules Santorico Page 98

2 Section 4-1: Sample Spaces and Probability Probability - the likelihood of an event occurring. Probability experiment a chance process that leads to welldefined results called outcomes. (i.e., some mechanism that produces a set of outcomes in a random way). Outcome the result of a single trial of a probability experiment. Example: Roll a die once. What could happen in one roll of the die? Ch4: Probability and Counting Rules Santorico Page 99

3 Sample space the set of all possible outcomes of a probability experiment. Example: What is the sample space for one flip of a coin? Heads, Tails Example: Suppose I roll two six-sided dice. What is the sample space for the possible outcomes? 1, 2, 3, 4, 5, 6 Ch4: Probability and Counting Rules Santorico Page 100

4 Example: Find the sample space for drawing one card from an ordinary deck of cards. Sample space consists of all possible 13x4=52 outcomes: A, 2,,K,, A, 2,,K Ch4: Probability and Counting Rules Santorico Page 101

5 TREE DIAGRAM a device consisting of line segments emanating from a starting point and also from the outcome points. It is used to determine all possible outcomes of a probability experiment. Example: Use a tree diagram to find the sample space for the sex of three children in a family. Our outcome pertains to the sex of one child AND the second of the next child AND the sex of the third child. Each of the children will correspond to a branching in the tree. What is the sex of the first child? Boy/Girl Given the sex of the first child, what is the sex of the second child? Given the sex of the first two children, what is the sex of the third child? Ch4: Probability and Counting Rules Santorico Page 102

6 Ch4: Probability and Counting Rules Santorico Page 103

7 Example: 3 pairs of jeans, 5 shirts, 2 hats. Use a tree diagram to determine all possible outfits composed of a pair of jeans, shirt, and a hat. Ch4: Probability and Counting Rules Santorico Page 104

8 Event consists of a set of possible outcomes of a probability experiment. Can be one outcome or more than one outcome. Simple event an event with one outcome. Compound event an event with more than one outcome. Example: Roll a die and get a 6 (simple event). Example: Roll a die and get an even number (compound event). Ch4: Probability and Counting Rules Santorico Page 105

9 There are three basic interpretations or probability: 1. Classical probability 2. Experimental or relative frequency probability 3. Subjective probability Theoretical (Classical) Probability uses sample spaces to determine the numerical probability that an event will happen. We do not actually perform the experiment to determine the theoretical probability. Assumes that all outcomes are equally likely to occur. Ch4: Probability and Counting Rules Santorico Page 106

10 Formula for Classic Probability The probability of an event E is P(E) Number of outcomes in E Number of outcomes in the sample space n(e) n(s) where S denotes the sample space and n( ) means the number of outcomes in... Rounding Rules for Probabilities probabilities should be expressed as reduced fractions or rounded to 2-3 decimal places. If the probability is extremely small then round to the first nonzero digit. Ch4: Probability and Counting Rules Santorico Page 107

11 Example: Consider a standard deck of 52 cards: Find the probability of selecting a queen 4 1 P queen CAN LEAVE AS A REDUCED FRACTION! This is to demonstrate rounding. Find the probability of selecting a spade, P(spade) = Find the probability of selecting a red ace, P(red ace) = Ch4: Probability and Counting Rules Santorico Page 108

12 Probability Rules 1. The Probability of an event E must be a number between 0 and 1. i.e., 0 P(E) If an event E cannot occur, then its probability is If an event E must occur, then its probability is The sum of all probabilities of all the outcomes in the sample space is 1. Always a good sanity check when doing calculations! Ch4: Probability and Counting Rules Santorico Page 109

13 Example: Suppose I roll a standard six-sided die. What s the probability I get a 7? What s the probability that I get a number less than 7? What s the probability that I get a 1 or a 2 or a 3 or a 4 or a 5 or a 6? Ch4: Probability and Counting Rules Santorico Page 110

14 Complementary Events Complement of an event E - the set of outcomes in the sample space that are not included in the outcomes of event E. The complement of E is denoted by E ( E bar ). Note: The outcomes of an event and the outcomes of the complement make up the entire sample space. Ch4: Probability and Counting Rules Santorico Page 111

15 Venn Diagram a visual way of representing probabilities. Venn diagrams are a wonderful tool to help think through probability calculations. Ch4: Probability and Counting Rules Santorico Page 112

16 Example: What is the complement of the following events? Rolling a six-sided die and getting a 4? Complement = Rolling a die and getting 1,2, 3, 5 or 6. Rolling a die and getting a multiple of 3? Selecting a day of the week and getting a weekday? Selecting a month and getting a month that begins with an A? Ch4: Probability and Counting Rules Santorico Page 113

17 Rule for Complementary Events: P(E) 1 P(E) or P(E) 1 P(E) or P(E) P(E ) 1. Example: The probability of purchasing a defective light bulb is 12%. What is the probability of not purchasing a defective light bulb? P(not defective) = 1 P(defective) = =0.88 Example: What is the probability of not selecting a club in a standard deck of 52 cards? Ch4: Probability and Counting Rules Santorico Page 114

18 Empirical Probability the relative frequency of an event occurring from a probability experiment over the long-run. It relies on actual experience to determine the likelihood of an outcome rather than assuming equally likely outcomes. Given a frequency distribution, the probability of an event being in a given class is: P(E) frequency for the class total frequencies in the distribution f n. This probability is called the empirical probability. Ch4: Probability and Counting Rules Santorico Page 115

19 Example: Observe the proportion of male babies out of many, many births. Example: Major Field of Study Class Frequency Math 5 History 7 English 4 Science 9 25 What is the probability of being a math major? Science major? History or English Major? P(M) = 5/25=0.2 P(S) = P(H or E) = Ch4: Probability and Counting Rules Santorico Page 116

20 The Law of Large Numbers tells us that the as the number of trials increases the empirical probability gets closer to the theoretical (true) probability. Because of the law of large numbers we will interpret the probability to be the long-run results (which we know approximates the theoretical probability). The probability of a particular outcome is the proportion of times the outcome would occur in a long-run of observations. Ch4: Probability and Counting Rules Santorico Page 117

21 Example: Proportion of times a fair coin comes up as a head Ch4: Probability and Counting Rules Santorico Page 118

22 Subjective Probability uses a probability value based on an educated guess or estimate, employing opinions and inexact information. Often, you cannot repeat the probability experiment. Example: What is the probability you will pass this class? Example: What is the probability that you will get a certain job when you apply? Read about Probability and Risk Taking on pg We are notoriously bad at subjectively estimating probability! Ch4: Probability and Counting Rules Santorico Page 119

23 SECTION 4-2: THE ADDITION RULES FOR PROBABILITY There are times when we want to find the probability of two or more events. For example, when selecting a card from a deck we may want to find the probability of selecting a card that is a four or red. In this case there are 3 possibilities to consider: The card is a four The card is red The card is a four and red Now consider selecting a card and we want to find the probability of selecting a card that is a spade or a diamond. In this case there are only 2 possibilities to consider: The card is a spade The card is a diamond Notice it can t be both a spade and a diamond. Ch4: Probability and Counting Rules Santorico Page 120

24 Mutually exclusive - Two events are mutually exclusive (disjoint) if they cannot occur at the same time. Looking ahead: If we have mutually exclusive events, then their probabilities will add. Let s make sure we understand what it means for events to be mutually exclusive. Example: Which events are mutually exclusive and which are not, when a single die is rolled? 1. Getting an odd number and getting an even number Mutually exclusive! You can t have a roll be both. 2. Getting a 3 and getting an odd number 3. Getting an odd number and getting a number less than 4 4. Getting a number greater than 4 and getting a number less than 4 Ch4: Probability and Counting Rules Santorico Page 121

25 Intersection the intersection of events A and B are the outcomes that are in both A and B. If A and B have outcomes intersecting each other than we say that they are nonmutually exclusive. Union the union of events A and B are all the outcomes that are in A, B, or both. Ch4: Probability and Counting Rules Santorico Page 122

26 Example: Suppose we roll a six-sided die. Let A be that we roll an even number. Let B be that we roll a number greater than 3. A A A B B B What is the intersection between A and B? Rolling a 6 or 4 What is the union of A and B? Rolling a 6, 5, 4, or 2 Ch4: Probability and Counting Rules Santorico Page 123

27 Addition Rules (These apply to or statements.) Rule 1: If two events A and B are mutually exclusive, then: P(A or B) = P(A) + P(B) Rule 2: For ANY two outcomes A and B, P(A or B) = P(A) + P(B) P(A and B) Note: In probability A or B denotes that A occurs, or B occurs, or both occur! Ch4: Probability and Counting Rules Santorico Page 124

28 Venn diagrams for mutually versus non-mutually exclusive events: Ch4: Probability and Counting Rules Santorico Page 125

29 Example: At a political rally, there are 20 Republicans, 13 Democrats, and 6 Independents. If a person is selected at random, find the probability that he or she is either a Democrat or an Independent. Event A = a person is a democrat Event B = a person is an independent These are mutually exclusive since you can NOT be both. P A P B P a person is a Democrat or an Independent P A or B Ch4: Probability and Counting Rules Santorico Page 126

30 Example: A single card is drawn at random from an ordinary deck of cards. Find the probability that the card is either an ace or a red card. (Hint: Define events. Determine if mutually exclusive. Use appropriate rule on slide 28.) Ch4: Probability and Counting Rules Santorico Page 127

31 Example: On New Year s Eve, the probability of a person driving while intoxicated is 0.32, the probability of a person having a driving accident is 0.09, and the probability of a person having a driving accident while intoxicated is What is the probability of a person driving while intoxicated or having a driving accident? Ch4: Probability and Counting Rules Santorico Page 128

32 Section 4-3 The Multiplication Rules and Conditional Probability The multiplication rules can be used to find the probabilities of two or more events that occur in sequence. When there are multiple stages in the experiment When there are 2 or more trials For example, tossing a coin then rolling a die (a 2-stage experiment) Multiplication rules apply to and statements. Ch4: Probability and Counting Rules Santorico Page 129

33 Independent - two events A and B are independent events if the fact that A occurs does not affect the probability of B occurring. Example: Rolling one die and getting a six, rolling a second die and getting a three. Example: Draw a card from a deck and replacing it, drawing a second card from the deck and getting a queen. In each example, the first event has no effect on the probability of the second event. Ch4: Probability and Counting Rules Santorico Page 130

34 Multiplication Rule for Independent Events Multiplication Rule 1: When two events A and B are independent, then P(A and B) P(A)P(B) That is, when events are independent, their probabilities multiply in an and statement. Example: The New York state lottery uses balls numbered 0-9 circulating in 3 separation bins. To select the winning sequence, one ball is chosen at random from each bin. What is the probability that the sequence would be the one selected? P(Sequence 9 1 1) Actually, this is the same probability of any of the equally likely 1000 draws Ch4: Probability and Counting Rules Santorico Page 131

35 Example: Approximately 9% of men have a type of color blindness that prevents them from distinguishing between red and green. If 3 men are selected at random, find the probability that all of them will have this type of red-green color blindness. Ch4: Probability and Counting Rules Santorico Page 132

36 Dependent - Two outcomes are said to be dependent if knowing that one of the outcomes has occurred affects the probability that the other occurs. Examples: Drawing a card from a deck, not replacing it, and then drawing a second card. Being a lifeguard and getting a suntan Having high grades and getting a scholarship Parking in a no-parking zone and getting a ticket Ch4: Probability and Counting Rules Santorico Page 133

37 The Conditional Probability of an event B in relationship to an event A is the probability that event B occurs after event A has already occurred. This probability is denoted as P(B A). When events are dependent we cannot use the current form of the multiplication rule. We have to modify it. Ch4: Probability and Counting Rules Santorico Page 134

38 Multiplication Rule for Dependent Events Multiplication Rule 2: When two events are dependent, the probability of both occurring is P(A and B) P(A)P(B A). Example: What is the probability of getting an Ace on the first draw and a king on a second draw? P(Ace then King)= P(Ace) P(King Ace) First draw from full deck of 52 cards has 4 Aces Second draw from a deck of 51 cards (which is missing a single Ace) has 4 Kings Ch4: Probability and Counting Rules Santorico Page 135

39 Example: At a university in western Pennsylvania, there were 5 burglaries reported in 2003, 16 in 2004, and 32 in If a researcher wishes to select two burglaries at random to further investigate, find the probability that both will have occurred in 2004? Ch4: Probability and Counting Rules Santorico Page 136

40 Example: World Wide Insurance Company found that 53% of the residents of a city had homeowner s insurance (H) with the company. Of these clients, 27% also had automobile insurance (A) with the company. If a resident is selected at random, find the probability that the resident has both homeowner s insurance and automobile insurance with World Wide Insurance Company. P( H and A) P( H) P( A H) Ch4: Probability and Counting Rules Santorico Page 137

41 Formula for Conditional Probability The probability that the second event B occurs given that the first event A has occurred can be found by dividing the probability that both events have occurred by the probability that the first event has occurred. For events A and B, the conditional probability of event B given A occurred is P(B A) P(A and B) P(A) Ch4: Probability and Counting Rules Santorico Page 138

42 Example: A box contains black chips and white chips. A person selects two chips without replacement. If the probability of selecting a black chip and a white chip is 15/56, and the probability of selecting a black chip in the first draw is 3/8, find the probability of selecting the white chip on the second draw, given that the first chip selected was a black chip. Want to compute: P(White chip on second draw First chip was black) Know: P(Selecting black and white chip)=15/56 P(Selecting black chip on first draw)=3/8 Applying formula for conditional probability: P(White chip on second draw First chip was black) P(Selecting black and white chip) 15 / 56 = P(First chip was black) 3/ 8 Ch4: Probability and Counting Rules Santorico Page 139

43 Example: The probability that Sam parks in a no-parking zone and gets a parking ticket is The probability that Sam cannot find a legal parking space and has to park in the noparking zone is Today, Sam arrived at UCD and has to park in a no-parking zone. Find the probability that he will get a parking ticket. Let N = parking in a no-parking zone, T = getting a ticket Ch4: Probability and Counting Rules Santorico Page 140

44 Venn Diagram for Conditional Probability Ch4: Probability and Counting Rules Santorico Page 141

45 Probabilities for At Least The multiplication rules can be coupled with the complimentary event rules (Section 4-1) to solve probability problems involving at least. Example: A game is played by drawing 4 cards from an ordinary deck and replacing each card after it is drawn. Find the probability that at least 1 ace is drawn. P at least 1 Ace 1 P no aces drawn Complementation Multiplication Rule Note rounding to 3 decimal places. Ch4: Probability and Counting Rules Santorico Page 142

46 What keyword in a probability problem probably means you should use the additional rules? What keyword in a probability problem probably means you should use the multiplication rules? Ch4: Probability and Counting Rules Santorico Page 143

47 Example: Consider a system of four components, as pictured in the diagram. Components 1 and 2 form a series subsystem, as do Components 3 and 4. The two subsystems are connected in parallel. Suppose that P(1 works)=.9, P(2 works)=.9, P(3 works)=.9, P(4 works)=.9, and that the four outcomes are independent of each other. Ch4: Probability and Counting Rules Santorico Page 144

48 The subsystem works only if both components work. What is the probability the 1-2 subsystem works? What is the probability that the 1-2 subsystem doesn t work? That the 3-4 subsystem doesn t work? The system won t work if both the 1-2 subsystem doesn t work and the 3-4 subsystem doesn t work. What is the probability that it won t work? That it will work? Ch4: Probability and Counting Rules Santorico Page 145

49 Ch4: Probability and Counting Rules Santorico Page 146

### Probability - Chapter 4

Probability - Chapter 4 In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. A cynical person

### Probability and Counting Rules. Chapter 3

Probability and Counting Rules Chapter 3 Probability as a general concept can be defined as the chance of an event occurring. Many people are familiar with probability from observing or playing games of

### 1. How to identify the sample space of a probability experiment and how to identify simple events

Statistics Chapter 3 Name: 3.1 Basic Concepts of Probability Learning objectives: 1. How to identify the sample space of a probability experiment and how to identify simple events 2. How to use the Fundamental

### Chapter 4. Probability and Counting Rules. McGraw-Hill, Bluman, 7 th ed, Chapter 4

Chapter 4 Probability and Counting Rules McGraw-Hill, Bluman, 7 th ed, Chapter 4 Chapter 4 Overview Introduction 4-1 Sample Spaces and Probability 4-2 Addition Rules for Probability 4-3 Multiplication

### 4.1 Sample Spaces and Events

4.1 Sample Spaces and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment is called an

### Probability. The MEnTe Program Math Enrichment through Technology. Title V East Los Angeles College

Probability The MEnTe Program Math Enrichment through Technology Title V East Los Angeles College 2003 East Los Angeles College. All rights reserved. Topics Introduction Empirical Probability Theoretical

### Chapter 1. Probability

Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.

### Probability as a general concept can be defined as the chance of an event occurring.

3. Probability In this chapter, you will learn about probability its meaning, how it is computed, and how to evaluate it in terms of the likelihood of an event actually happening. Probability as a general

### Probability Concepts and Counting Rules

Probability Concepts and Counting Rules Chapter 4 McGraw-Hill/Irwin Dr. Ateq Ahmed Al-Ghamedi Department of Statistics P O Box 80203 King Abdulaziz University Jeddah 21589, Saudi Arabia ateq@kau.edu.sa

### Math 1313 Section 6.2 Definition of Probability

Math 1313 Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability

### CHAPTERS 14 & 15 PROBABILITY STAT 203

CHAPTERS 14 & 15 PROBABILITY STAT 203 Where this fits in 2 Up to now, we ve mostly discussed how to handle data (descriptive statistics) and how to collect data. Regression has been the only form of statistical

### Chapter 5 - Elementary Probability Theory

Chapter 5 - Elementary Probability Theory Historical Background Much of the early work in probability concerned games and gambling. One of the first to apply probability to matters other than gambling

### Empirical (or statistical) probability) is based on. The empirical probability of an event E is the frequency of event E.

Probability and Statistics Chapter 3 Notes Section 3-1 I. Probability Experiments. A. When weather forecasters say There is a 90% chance of rain tomorrow, or a doctor says There is a 35% chance of a successful

### Chapter 1. Probability

Chapter 1. Probability 1.1 Basic Concepts Scientific method a. For a given problem, we define measures that explains the problem well. b. Data is collected with observation and the measures are calculated.

### Math 227 Elementary Statistics. Bluman 5 th edition

Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 4 Probability and Counting Rules 2 Objectives Determine sample spaces and find the probability of an event using classical probability or empirical

### Probability. Ms. Weinstein Probability & Statistics

Probability Ms. Weinstein Probability & Statistics Definitions Sample Space The sample space, S, of a random phenomenon is the set of all possible outcomes. Event An event is a set of outcomes of a random

### 4.3 Rules of Probability

4.3 Rules of Probability If a probability distribution is not uniform, to find the probability of a given event, add up the probabilities of all the individual outcomes that make up the event. Example:

### 7.1 Experiments, Sample Spaces, and Events

7.1 Experiments, Sample Spaces, and Events An experiment is an activity that has observable results. Examples: Tossing a coin, rolling dice, picking marbles out of a jar, etc. The result of an experiment

### Chapter 5 Probability

Chapter 5 Probability Math150 What s the likelihood of something occurring? Can we answer questions about probabilities using data or experiments? For instance: 1) If my parking meter expires, I will probably

### Section Introduction to Sets

Section 1.1 - Introduction to Sets Definition: A set is a well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase

### Such a description is the basis for a probability model. Here is the basic vocabulary we use.

5.2.1 Probability Models When we toss a coin, we can t know the outcome in advance. What do we know? We are willing to say that the outcome will be either heads or tails. We believe that each of these

### ECON 214 Elements of Statistics for Economists

ECON 214 Elements of Statistics for Economists Session 4 Probability Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education School of Continuing

### Probability Quiz Review Sections

CP1 Math 2 Unit 9: Probability: Day 7/8 Topic Outline: Probability Quiz Review Sections 5.02-5.04 Name A probability cannot exceed 1. We express probability as a fraction, decimal, or percent. Probabilities

### Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1 - Experiments, Sample Spaces,

### Math Exam 2 Review. NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5.

Math 166 Spring 2007 c Heather Ramsey Page 1 Math 166 - Exam 2 Review NOTE: For reviews of the other sections on Exam 2, refer to the first page of WIR #4 and #5. Section 7.1 - Experiments, Sample Spaces,

### Intermediate Math Circles November 1, 2017 Probability I

Intermediate Math Circles November 1, 2017 Probability I Probability is the study of uncertain events or outcomes. Games of chance that involve rolling dice or dealing cards are one obvious area of application.

### Def: The intersection of A and B is the set of all elements common to both set A and set B

Def: Sample Space the set of all possible outcomes Def: Element an item in the set Ex: The number "3" is an element of the "rolling a die" sample space Main concept write in Interactive Notebook Intersection:

### Probability. Probabilty Impossibe Unlikely Equally Likely Likely Certain

PROBABILITY Probability The likelihood or chance of an event occurring If an event is IMPOSSIBLE its probability is ZERO If an event is CERTAIN its probability is ONE So all probabilities lie between 0

### If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics

If you roll a die, what is the probability you get a four OR a five? What is the General Education Statistics probability that you get neither? Class Notes The Addition Rule (for OR events) and Complements

### PROBABILITY. 1. Introduction. Candidates should able to:

PROBABILITY Candidates should able to: evaluate probabilities in simple cases by means of enumeration of equiprobable elementary events (e.g for the total score when two fair dice are thrown), or by calculation

### Probability and Randomness. Day 1

Probability and Randomness Day 1 Randomness and Probability The mathematics of chance is called. The probability of any outcome of a chance process is a number between that describes the proportion of

### Business Statistics. Chapter 4 Using Probability and Probability Distributions QMIS 120. Dr. Mohammad Zainal

Department of Quantitative Methods & Information Systems Business Statistics Chapter 4 Using Probability and Probability Distributions QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter,

### Here are other examples of independent events:

5 The Multiplication Rules and Conditional Probability The Multiplication Rules Objective. Find the probability of compound events using the multiplication rules. The previous section showed that the addition

### Probability Rules. 2) The probability, P, of any event ranges from which of the following?

Name: WORKSHEET : Date: Answer the following questions. 1) Probability of event E occurring is... P(E) = Number of ways to get E/Total number of outcomes possible in S, the sample space....if. 2) The probability,

### Grade 6 Math Circles Fall Oct 14/15 Probability

1 Faculty of Mathematics Waterloo, Ontario Centre for Education in Mathematics and Computing Grade 6 Math Circles Fall 2014 - Oct 14/15 Probability Probability is the likelihood of an event occurring.

### 1MA01: Probability. Sinéad Ryan. November 12, 2013 TCD

1MA01: Probability Sinéad Ryan TCD November 12, 2013 Definitions and Notation EVENT: a set possible outcomes of an experiment. Eg flipping a coin is the experiment, landing on heads is the event If an

### MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Study Guide for Test III (MATH 1630) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the number of subsets of the set. 1) {x x is an even

### Chapter 3: Probability (Part 1)

Chapter 3: Probability (Part 1) 3.1: Basic Concepts of Probability and Counting Types of Probability There are at least three different types of probability Subjective Probability is found through people

### Textbook: pp Chapter 2: Probability Concepts and Applications

1 Textbook: pp. 39-80 Chapter 2: Probability Concepts and Applications 2 Learning Objectives After completing this chapter, students will be able to: Understand the basic foundations of probability analysis.

### Probability is the likelihood that an event will occur.

Section 3.1 Basic Concepts of is the likelihood that an event will occur. In Chapters 3 and 4, we will discuss basic concepts of probability and find the probability of a given event occurring. Our main

### Probability Models. Section 6.2

Probability Models Section 6.2 The Language of Probability What is random? Empirical means that it is based on observation rather than theorizing. Probability describes what happens in MANY trials. Example

### Define and Diagram Outcomes (Subsets) of the Sample Space (Universal Set)

12.3 and 12.4 Notes Geometry 1 Diagramming the Sample Space using Venn Diagrams A sample space represents all things that could occur for a given event. In set theory language this would be known as the

### Grade 7/8 Math Circles February 25/26, Probability

Faculty of Mathematics Waterloo, Ontario N2L 3G1 Probability Grade 7/8 Math Circles February 25/26, 2014 Probability Centre for Education in Mathematics and Computing Probability is the study of how likely

### The Teachers Circle Mar. 20, 2012 HOW TO GAMBLE IF YOU MUST (I ll bet you \$5 that if you give me \$10, I ll give you \$20.)

The Teachers Circle Mar. 2, 22 HOW TO GAMBLE IF YOU MUST (I ll bet you \$ that if you give me \$, I ll give you \$2.) Instructor: Paul Zeitz (zeitzp@usfca.edu) Basic Laws and Definitions of Probability If

### November 6, Chapter 8: Probability: The Mathematics of Chance

Chapter 8: Probability: The Mathematics of Chance November 6, 2013 Last Time Crystallographic notation Groups Crystallographic notation The first symbol is always a p, which indicates that the pattern

### "Well, statistically speaking, you are for more likely to have an accident at an intersection, so I just make sure that I spend less time there.

6.2 Probability Models There was a statistician who, when driving his car, would always accelerate hard before coming to an intersection, whiz straight through it, and slow down again once he was beyond

### Independent and Mutually Exclusive Events

Independent and Mutually Exclusive Events By: OpenStaxCollege Independent and mutually exclusive do not mean the same thing. Independent Events Two events are independent if the following are true: P(A

### Probability Review 41

Probability Review 41 For the following problems, give the probability to four decimals, or give a fraction, or if necessary, use scientific notation. Use P(A) = 1 - P(not A) 1) A coin is tossed 6 times.

### Probability MAT230. Fall Discrete Mathematics. MAT230 (Discrete Math) Probability Fall / 37

Probability MAT230 Discrete Mathematics Fall 2018 MAT230 (Discrete Math) Probability Fall 2018 1 / 37 Outline 1 Discrete Probability 2 Sum and Product Rules for Probability 3 Expected Value MAT230 (Discrete

### CHAPTER 7 Probability

CHAPTER 7 Probability 7.1. Sets A set is a well-defined collection of distinct objects. Welldefined means that we can determine whether an object is an element of a set or not. Distinct means that we can

### Module 4 Project Maths Development Team Draft (Version 2)

5 Week Modular Course in Statistics & Probability Strand 1 Module 4 Set Theory and Probability It is often said that the three basic rules of probability are: 1. Draw a picture 2. Draw a picture 3. Draw

### Chapter 3: PROBABILITY

Chapter 3 Math 3201 1 3.1 Exploring Probability: P(event) = Chapter 3: PROBABILITY number of outcomes favourable to the event total number of outcomes in the sample space An event is any collection of

### Key Concepts. Theoretical Probability. Terminology. Lesson 11-1

Key Concepts Theoretical Probability Lesson - Objective Teach students the terminology used in probability theory, and how to make calculations pertaining to experiments where all outcomes are equally

### Probability Test Review Math 2. a. What is? b. What is? c. ( ) d. ( )

Probability Test Review Math 2 Name 1. Use the following venn diagram to answer the question: Event A: Odd Numbers Event B: Numbers greater than 10 a. What is? b. What is? c. ( ) d. ( ) 2. In Jason's homeroom

### I. WHAT IS PROBABILITY?

C HAPTER 3 PROAILITY Random Experiments I. WHAT IS PROAILITY? The weatherman on 10 o clock news program states that there is a 20% chance that it will snow tomorrow, a 65% chance that it will rain and

### Class XII Chapter 13 Probability Maths. Exercise 13.1

Exercise 13.1 Question 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E F) = 0.2, find P (E F) and P(F E). It is given that P(E) = 0.6, P(F) = 0.3, and P(E F) = 0.2 Question 2:

### Raise your hand if you rode a bus within the past month. Record the number of raised hands.

166 CHAPTER 3 PROBABILITY TOPICS Raise your hand if you rode a bus within the past month. Record the number of raised hands. Raise your hand if you answered "yes" to BOTH of the first two questions. Record

### Chapter 1: Sets and Probability

Chapter 1: Sets and Probability Section 1.3-1.5 Recap: Sample Spaces and Events An is an activity that has observable results. An is the result of an experiment. Example 1 Examples of experiments: Flipping

### CHAPTER 2 PROBABILITY. 2.1 Sample Space. 2.2 Events

CHAPTER 2 PROBABILITY 2.1 Sample Space A probability model consists of the sample space and the way to assign probabilities. Sample space & sample point The sample space S, is the set of all possible outcomes

### Chapter 11: Probability and Counting Techniques

Chapter 11: Probability and Counting Techniques Diana Pell Section 11.3: Basic Concepts of Probability Definition 1. A sample space is a set of all possible outcomes of an experiment. Exercise 1. An experiment

### Exercise Class XI Chapter 16 Probability Maths

Exercise 16.1 Question 1: Describe the sample space for the indicated experiment: A coin is tossed three times. A coin has two faces: head (H) and tail (T). When a coin is tossed three times, the total

### Chapter 5: Probability: What are the Chances? Section 5.2 Probability Rules

+ Chapter 5: Probability: What are the Chances? Section 5.2 + Two-Way Tables and Probability When finding probabilities involving two events, a two-way table can display the sample space in a way that

### A Probability Work Sheet

A Probability Work Sheet October 19, 2006 Introduction: Rolling a Die Suppose Geoff is given a fair six-sided die, which he rolls. What are the chances he rolls a six? In order to solve this problem, we

### Introduction to Probability and Statistics I Lecture 7 and 8

Introduction to Probability and Statistics I Lecture 7 and 8 Basic Probability and Counting Methods Computing theoretical probabilities:counting methods Great for gambling! Fun to compute! If outcomes

### 5 Elementary Probability Theory

5 Elementary Probability Theory 5.1 What is Probability? The Basics We begin by defining some terms. Random Experiment: any activity with a random (unpredictable) result that can be measured. Trial: one

### Lecture 6 Probability

Lecture 6 Probability Example: When you toss a coin, there are only two possible outcomes, heads and tails. What if we toss a coin two times? Figure below shows the results of tossing a coin 5000 times

### Total. STAT/MATH 394 A - Autumn Quarter Midterm. Name: Student ID Number: Directions. Complete all questions.

STAT/MATH 9 A - Autumn Quarter 015 - Midterm Name: Student ID Number: Problem 1 5 Total Points Directions. Complete all questions. You may use a scientific calculator during this examination; graphing

### PROBABILITY. Example 1 The probability of choosing a heart from a deck of cards is given by

Classical Definition of Probability PROBABILITY Probability is the measure of how likely an event is. An experiment is a situation involving chance or probability that leads to results called outcomes.

### Probability and Counting Techniques

Probability and Counting Techniques Diana Pell (Multiplication Principle) Suppose that a task consists of t choices performed consecutively. Suppose that choice 1 can be performed in m 1 ways; for each

### Week 3 Classical Probability, Part I

Week 3 Classical Probability, Part I Week 3 Objectives Proper understanding of common statistical practices such as confidence intervals and hypothesis testing requires some familiarity with probability

### Classical vs. Empirical Probability Activity

Name: Date: Hour : Classical vs. Empirical Probability Activity (100 Formative Points) For this activity, you will be taking part in 5 different probability experiments: Rolling dice, drawing cards, drawing

### Quiz 2 Review - on Notebook Paper Are You Ready For Your Last Quiz In Honors Math II??

Quiz 2 Review - on Notebook Paper Are You Ready For Your Last Quiz In Honors Math II?? Some things to Know, Memorize, AND Understand how to use are n What are the formulas? Pr ncr Fill in the notation

### 8.2 Union, Intersection, and Complement of Events; Odds

8.2 Union, Intersection, and Complement of Events; Odds Since we defined an event as a subset of a sample space it is natural to consider set operations like union, intersection or complement in the context

### MATH STUDENT BOOK. 7th Grade Unit 6

MATH STUDENT BOOK 7th Grade Unit 6 Unit 6 Probability and Graphing Math 706 Probability and Graphing Introduction 3 1. Probability 5 Theoretical Probability 5 Experimental Probability 13 Sample Space 20

### Contents 2.1 Basic Concepts of Probability Methods of Assigning Probabilities Principle of Counting - Permutation and Combination 39

CHAPTER 2 PROBABILITY Contents 2.1 Basic Concepts of Probability 38 2.2 Probability of an Event 39 2.3 Methods of Assigning Probabilities 39 2.4 Principle of Counting - Permutation and Combination 39 2.5

### Unit 7 Central Tendency and Probability

Name: Block: 7.1 Central Tendency 7.2 Introduction to Probability 7.3 Independent Events 7.4 Dependent Events 7.1 Central Tendency A central tendency is a central or value in a data set. We will look at

### 3 The multiplication rule/miscellaneous counting problems

Practice for Exam 1 1 Axioms of probability, disjoint and independent events 1. Suppose P (A) = 0.4, P (B) = 0.5. (a) If A and B are independent, what is P (A B)? What is P (A B)? (b) If A and B are disjoint,

### Chapter 6: Probability and Simulation. The study of randomness

Chapter 6: Probability and Simulation The study of randomness Introduction Probability is the study of chance. 6.1 focuses on simulation since actual observations are often not feasible. When we produce

### Chapter 6: Probability and Simulation. The study of randomness

Chapter 6: Probability and Simulation The study of randomness 6.1 Randomness Probability describes the pattern of chance outcomes. Probability is the basis of inference Meaning, the pattern of chance outcomes

### Basic Concepts of Probability and Counting Section 3.1

Basic Concepts of Probability and Counting Section 3.1 Summer 2013 - Math 1040 June 17 (1040) M 1040-3.1 June 17 1 / 12 Roadmap Basic Concepts of Probability and Counting Pages 128-137 Counting events,

### Section 6.5 Conditional Probability

Section 6.5 Conditional Probability Example 1: An urn contains 5 green marbles and 7 black marbles. Two marbles are drawn in succession and without replacement from the urn. a) What is the probability

### , -the of all of a probability experiment. consists of outcomes. (b) List the elements of the event consisting of a number that is greater than 4.

4-1 Sample Spaces and Probability as a general concept can be defined as the chance of an event occurring. In addition to being used in games of chance, probability is used in the fields of,, and forecasting,

### Mutually Exclusive Events

Mutually Exclusive Events Suppose you are rolling a six-sided die. What is the probability that you roll an odd number and you roll a 2? Can these both occur at the same time? Why or why not? Mutually

### Unit 1 Day 1: Sample Spaces and Subsets. Define: Sample Space. Define: Intersection of two sets (A B) Define: Union of two sets (A B)

Unit 1 Day 1: Sample Spaces and Subsets Students will be able to (SWBAT) describe events as subsets of sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions,

### Objective 1: Simple Probability

Objective : Simple Probability To find the probability of event E, P(E) number of ways event E can occur total number of outcomes in sample space Example : In a pet store, there are 5 puppies, 22 kittens,

### Probability - Grade 10 *

OpenStax-CNX module: m32623 1 Probability - Grade 10 * Rory Adams Free High School Science Texts Project Sarah Blyth Heather Williams This work is produced by OpenStax-CNX and licensed under the Creative

### Review. Natural Numbers: Whole Numbers: Integers: Rational Numbers: Outline Sec Comparing Rational Numbers

FOUNDATIONS Outline Sec. 3-1 Gallo Name: Date: Review Natural Numbers: Whole Numbers: Integers: Rational Numbers: Comparing Rational Numbers Fractions: A way of representing a division of a whole into

### Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results:

Lenarz Math 102 Practice Exam # 3 Name: 1. A 10-sided die is rolled 100 times with the following results: Outcome Frequency 1 8 2 8 3 12 4 7 5 15 8 7 8 8 13 9 9 10 12 (a) What is the experimental probability

### Name: Class: Date: 6. An event occurs, on average, every 6 out of 17 times during a simulation. The experimental probability of this event is 11

Class: Date: Sample Mastery # Multiple Choice Identify the choice that best completes the statement or answers the question.. One repetition of an experiment is known as a(n) random variable expected value

### Applications of Probability

Applications of Probability CK-12 Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive

### CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY

CHAPTER 9 - COUNTING PRINCIPLES AND PROBABILITY Probability is the Probability is used in many real-world fields, such as insurance, medical research, law enforcement, and political science. Objectives:

### When a number cube is rolled once, the possible numbers that could show face up are

C3 Chapter 12 Understanding Probability Essential question: How can you describe the likelihood of an event? Example 1 Likelihood of an Event When a number cube is rolled once, the possible numbers that

### 0-5 Adding Probabilities. 1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins.

1. CARNIVAL GAMES A spinner has sections of equal size. The table shows the results of several spins. d. a. Copy the table and add a column to show the experimental probability of the spinner landing on

### Chapter 3: Elements of Chance: Probability Methods

Chapter 3: Elements of Chance: Methods Department of Mathematics Izmir University of Economics Week 3-4 2014-2015 Introduction In this chapter we will focus on the definitions of random experiment, outcome,

### RANDOM EXPERIMENTS AND EVENTS

Random Experiments and Events 18 RANDOM EXPERIMENTS AND EVENTS In day-to-day life we see that before commencement of a cricket match two captains go for a toss. Tossing of a coin is an activity and getting

### North Seattle Community College Winter ELEMENTARY STATISTICS 2617 MATH Section 05, Practice Questions for Test 2 Chapter 3 and 4

North Seattle Community College Winter 2012 ELEMENTARY STATISTICS 2617 MATH 109 - Section 05, Practice Questions for Test 2 Chapter 3 and 4 1. Classify each statement as an example of empirical probability,

### Georgia Department of Education Georgia Standards of Excellence Framework GSE Geometry Unit 6

How Odd? Standards Addressed in this Task MGSE9-12.S.CP.1 Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events (or, and, not). MGSE9-12.S.CP.7